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Conservation of momentum, energy and an inelastic collision.

  1. Feb 21, 2012 #1
    From conservation of momentum and energy, it turns out that if two objects bump into each other and end up moving at the same speed, energy must have been lost. I know the formulas and calculations, but what physically happens at the contact site? Why cannot one object just push the other one, until it moves as fast as the first object, and then stop pushing? Why does it have to cause heat? How does it cause the heat?

    Another related question I have, is that why is momentum conserved, if energy is not, ie if some of the force doesn't get applied to the other body, but instead gets converted to heat, why is it that momentum is conserved? Conservation of momentum comes from action-reaction, if you push one object, you get pushed in the other direction with the same force, but if some of the force through distance (energy) is applied to heating, why is momentum still conserved?
  2. jcsd
  3. Feb 21, 2012 #2


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    The deformation must change if the force between the bodies changes. So even two perfectly elastic balls, which stick together after collision, will be "wobbling" afterwards. Some energy will be stored that way.
    In the real world deformation always causes dissipation through heat. It is the imperfection of the structures.

    Momentum is a vector. Energy is a scalar. Force through distance on two parts applied in opposite directions, doesn't change the total momentum.
  4. Feb 21, 2012 #3
    How is that heat generated when one object hits another stationary object? I guess the force pushes the molecular structure in, the structure doesn't restore it's original position, the added kinetic energy to the deformed structure stays within it and represents heat. What makes it necessary to happen (heat generation) when they stick together and wobbling and such isn't significant? Why cannot it just push the other one through a certain distance (transfer energy) and be done with it?
  5. Feb 21, 2012 #4
    Do you think the energy change in an inelastic collision is a loss due to heat? I don't think it is because if it were, we could measure the heat, subtract it from the original total, and find energy was conserved.
  6. Feb 21, 2012 #5


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    Energy is conserved, but mechanical energy is not conserved.

    It is easy to show that some of the energy is transformed into heat. Screw a metal screw half way into a wood block. Then try to hammer it further into the block as if it was a nail not a screw. You want to set this up so you can give it repeated hard blows (which are approximately inelastic collisions), but you don't move it. A big woodscrew, e.g. 3 or 4 inches long, works better than a small one.

    Then feel the temperature of the screw before it has time to cool down - and be careful you don't burn your fingers.
  7. Feb 21, 2012 #6
  8. Feb 21, 2012 #7
    Show us an explicit calculation which demonstrates the kinetic energy is lost.
  9. Feb 22, 2012 #8

    Doc Al

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    Do you doubt that kinetic energy is 'lost' in a perfectly inelastic collision?
  10. Feb 22, 2012 #9
    Object 1 with mass m and velocity v hits stationary object 2 with mass m and velocity=0. If this is inelastic (say the objects are 2 balls of putty) then the combined object has mass 2m and moves off with velocity v/2 (by conservation of momentum). Now check KE: before collision KE=(1/2)mv^2. After collision KE= (1/2)(2m)(v/2)^2=(1/4)mv^2. So, one-half of the initial KE has been lost by deforming & heating the putty balls.

    It's impressive that we can calculate the lost KE without knowing much about the nature of the balls, isn't it? I think that's the motivation for the OP's question.
  11. Feb 22, 2012 #10
    Yeah, this Ek wasn't lost to heat.
  12. Feb 22, 2012 #11
    No, but it's certainly not lost to heat.
  13. Feb 22, 2012 #12
    You asked:
    "Why cannot one object just push the other one, until it moves as fast as the first object, and then stop pushing? Why does it have to cause heat? How does it cause the heat?"

    I think this explanation helps:

    When two elastic objects collide, from the time they touch one another, they start being compressed with their center of mass keep getting closer to one another. There is a moment that the centers don't get closer, and that means their velocity becomes the same ( in that direction). But this moment happens after the objects has deformed, hence some energy has been stored in the deformation process. From that moment the force due to the elasticity pushes the objects in different directions till they lose contact. during these times their velocity deviates from the equal one they just had.

    If the two separate with the same speed, this means they separate at the moment of the maximum deformation which means the collision is not elastic and in an inelastic collision, mechanical energy is lost.
  14. Feb 22, 2012 #13
    The KE goes into deformation, vibration, noise, and heat. If the objects are simple (rigid point particles like say gas atoms) then there is no 'mechanism' to transform the KE into any other form; in that case though, there is no way to have an inelastic collision. It wouldn't be possible for the two objects to have the same speed after the collision (i.e., they wouldn't 'move off together').
  15. Feb 22, 2012 #14

    Doc Al

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    Why do you say that? (Of course, 'heat' is not the only form of energy that the KE is lost to.)
  16. Feb 22, 2012 #15
    Thank you! Exactly the explanation I was looking for. You made me now understand it: If one object is going faster than the other, it must be pushing into the other, causing deformation. If at a point of time during the collision, it is still going faster, it will still push into the other and cause deformation. When the velocities are equal, it is the point of maximum deformation, and depending on whether that deformation pushes back elastically or not or by how much determines how much energy is conserved in the kinetic energies of the objects and how much is stored in the deformation!
  17. Feb 22, 2012 #16
    The drop by 1/2 of the kinetic energy is due to the work of pushing the 2nd mass.

    In the momentum equation the cut in velocity is due to the increase in mass. The momentum is redistributed to twice the mass.

    Recalculating after the collision therefore, half the original velocity is plugged in to the energy equation. From this perspective, the energy went into the work of pushing the second particle from 0 velocity to the final velocity, F X d. Energy is the ability to do work. This energy did some work. A mass was moved. The heat created was almost certainly negligible by comparison to the work. Whatever increased thermal energy the 2nd particle may now have, it's change in kinetic energy from the former amount of 0 is quite a bit more salient.
    Last edited: Feb 22, 2012
  18. Feb 22, 2012 #17

    Doc Al

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    And where do you think the energy involved in the work done ends up? "Goes into work" is not an answer!
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